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Table 2 Comparison between Parkes’s (Parkes 2010b ) solutions and our solutions

From: Investigation of Solitary wave solutions for Vakhnenko-Parkes equation via exp-function and Exp(−ϕ(ξ))-expansion method

Parkes’s (Parkes2010b) Our solution
(i) If k2 = μ, η = ξ + C then solution u14 we obtain u ξ =−6μ−6μ cot 2 μ ξ + C (i) If λ = 0, then our solutions (33) reduced to u ξ =−6μ−6μ cot 2 μ ξ + C
(ii) If k2 = − μ, η = ξ + C then from solution u12 we obtain u ξ =−6μ+6μ coth 2 − μ ξ + C (ii) If λ = 0, then our solutions (28) reduced to u ξ =−6μ+6μ coth 2 − μ ξ + C
(iii) If k = λ/2, η = ξ + C then solution u12 we obtain u ξ = 3 2 λ 2 − 3 2 λ 2 coth 2 1 2 λ ξ + C (iii) Eq. (34) can be simplified to gives u ξ = 3 2 λ 2 − 3 2 λ 2 coth 2 1 2 λ ξ + C
(iv) If η = ξ + C + λ/2 then solution u3 we obtain u ξ =− 6 ξ + C + λ / 2 2 (iv) Eq. (35) can be simplified to gives u ξ =− 6 ξ + C + λ / 2 2
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